Simplify and expand the following expression: $ \dfrac{3n + 7}{n + 8}+\dfrac{4n - 10}{5n + 6} $
In order to add expressions, they must have a common denominator. Get both fractions over a common denominator of $(n + 8)(5n + 6)$ Multiply the first term by $\dfrac{5n + 6}{5n + 6}$ $ \begin{align*} \dfrac{3n + 7}{n + 8} \times \dfrac{5n + 6}{5n + 6} & = \dfrac{(3n + 7)(5n + 6)}{(n + 8)(5n + 6)} \\ & = \dfrac{15n^2 + 53n + 42}{(n + 8)(5n + 6)}\end{align*} $ Multiply the second term by $\dfrac{n + 8}{n + 8}$ $ \begin{align*} \dfrac{4n - 10}{5n + 6} \times \dfrac{n + 8}{n + 8} & = \dfrac{(4n - 10)(n + 8)}{(5n + 6)(n + 8)} \\ & = \dfrac{4n^2 + 22n - 80}{(5n + 6)(n + 8)}\end{align*} $ Now we have: $ = \dfrac{15n^2 + 53n + 42}{(n + 8)(5n + 6)} + \dfrac{4n^2 + 22n - 80}{(5n + 6)(n + 8)} $ Now both terms have a common denominator we can simply add the numerators: $ = \dfrac{15n^2 + 53n + 42 + 4n^2 + 22n - 80}{(n + 8)(5n + 6)} $ $ = \dfrac{19n^2 + 75n - 38}{(n + 8)(5n + 6)}$ Expand the denominator: $ = \dfrac{19n^2 + 75n - 38}{5n^2 + 46n + 48}$